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 \begin{document}
% The file aaai.sty is the style file for AAAI Press 
% proceedings, working notes, and technical reports.
%
\title{Symbolic Dynamic Programming for Continuous State and Action MDPs}
\author{Anonymous}
%\author{Zahra Zamani\\
%NICTA \& the ANU\\
%Canberra, Australia\\
%{\tt zahra.zamani@anu.edu.au}
%\And
%Scott Sanner\\
%NICTA \& the ANU\\
%Canberra, Australia\\
%{\tt ssanner@nicta.com.au}
%\And
%Cheng Feng\\
%Department of Aeronautics and\\
%Astronautics, MIT, USA\\
%{\tt cfang@mit.edu}
%}
\maketitle
\begin{abstract}
\begin{quote}
Many real-world decision-theoretic planning problems are naturally
modeled using both continuous state and action (CSA) spaces, yet
little work has provided \emph{exact} solutions for the case of
continuous actions.  In this work, we propose a symbolic dynamic
programming (SDP) solution to obtain the \emph{optimal closed-form}
value function and policy for CSA-MDPs with multivariate continuous
state \emph{and} actions, discrete noise, piecewise linear dynamics,
and piecewise linear (or restricted piecewise quadratic) reward.  Our
key contribution over previous SDP work is to show how the continuous
action maximization step in the dynamic programming backup can be
evaluated optimally and symbolically --- a task which amounts to
\emph{symbolic} constrained optimization subject to unknown state
parameters; we further integrate this technique to work with an
efficient and compact data structure for SDP --- the extended
algebraic decision diagram (XADD).  We demonstrate empirical results
on a didactic nonlinear planning example and two domains from operations
research to show the \emph{first automated exact solution} to these
problems.
\end{quote}
\end{abstract}

\section{Introduction}

\input introduction.tex

\section{Continuous State and Action MDPs}

\input csamdp.tex

\section{Symbolic Dynamic Programming (SDP)}

\input sdp.tex

\vspace{-3mm}
\section{Empirical Results}

\input empirical.tex

\vspace{-1mm}
\section{Related Work and Concluding Remarks}

\input related_work.tex

%\section{Concluding Remarks}

\input conclusion.tex

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